On Stability of Regional Orthomodular Posets

The set of regions of a transition system, ordered by set inclusion, is an orthomodular poset, often referred to as quantum logic, here called regional logic. Regional logics, which are known to be regular and rich, are the main subject of investigation in this work. Given a regular, rich logic L, one can build a transition system A, such that L embeds into the regional logic of A. Call a logic stable if the embedding is an isomorphism. We give some necessary conditions for a logic to be stable, and show that under these, the embedding has some stronger property. In particular, we show that any \(\{0,1\}\)-pasting of n stable logics is stable, and that, whenever L contains n maximal Boolean sublogics with pairwise identical intersections, L is stable. The full characterization of the class of stable logics is still an open problem.